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Presenter: Jen Hua ChiAdviser: Yeong Sung Lin
Network Games with Many Attackers and Defenders
Agenda
Introduction Network Games with Many
Defenders New Strategic Model
3
Introduction
Mavronicholas et al. started a line of research of network games. (2008)
Showing that no game with the defender playing a single edge has a pure Nash equilibrium unless it is a trivial graph.
1. Using bipartite graph2. Improving non-deterministic algorithm into a deterministic polynomial-time algorithm.
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Introduction
Theorem if G contains more than one edges,
then the game has no pure Nash Equilibrium
for any graph G, the game contains a matching mixed Nash equilibrium if and only if the vertices of the graph G can be partitioned into two sets A, B, such that A is an independent set of G and B is a vertex cover of the graph
Introduction
Definition: matching M a set M ⊆ E is a matching of G if no two edges in M share a vertex vertex cover a set V’ ⊆ V such that for every edge (u, v) ∈ E either u ∈ V or v ∈ V’ edge cover a set E’ ⊆ E such that for every vertex v ∈ V , there is an edge (v, u) ∈ E’
Introduction
Definition: a mixed strategy for player i ∈ N is a
probability distribution over its pure strategy set Si
edge model : defender protects a single link of the network
How to determine a Nash equilibrium
For n players: n players corresponds to each n-tuple of pure
strategies, one strategy being taken for each player. any n-tuple of strategies, may be regarded as a point in
the product space obtained by multiplying the n strategy spaces.
one such n-tuple counters another if the strategy of each player in the countering n-tuple yields the highest obtainable expectation for its player against, the n - 1 strategies of the other players in the countered n-tuple.
a self-countering n-tuple is called an equilibrium point.
How to determine an equilibrium point : About countering
The correspondence of each n-tuple with its set of countering n-tuples gives a one-to-many mapping of the product space into itself.
The set of countering points of a point is convex.
The closeness is equivalent to saying: if P1, P2, ... and Q1, Q2, ..., Qn, ... are sequences of points in the product space where Qn Q,
Pn P and Qn counters Pn then Q counters P
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How to determine an equilibrium point
Inferring from Kakutani's theorem that the mapping has a fixed point (i.e. point contained in its image). Hence there is an equilibrium point.
Kakutani’s theorem: It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it.
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Introduction : Motivation
According to Mavronicholas et al. (2006) research the existence problem of pure Nash
equilibria is solvable in polynomial time provided a graph-theoretic
characterization of mixed Nash equilibria introduced k-matching configurations that generalize matching configurations
provide a characterization of graphs admitting k-matching Nash equilibria
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Introduction : Motivation
According to Mavronicholas et al. (2006) research.. develop a polynomial-time algorithm for
computing k-matching Nash equilibria on graphs and exhibit the applicability of the algorithm for bipartite graphs
establish that the increased power of the defender results in an improved quality of protection of the network
obtain that the gain of the defender, which amounts to the expected number of the arrested harmful procedures, is linear to the parameter k
Network Games with Many Defenders
Undirected graph G = (V, E) vertex cover CV ⊆ V edge cover CE ⊆ E matching M: m (size) = |M| ≥ |M’| independent set IV ⊆ V if v is incident to an edge e: v ∈ e number of vertices: nV
number of edges: nE
Network Games with Many Defenders
Characterization of (pure) Nash equilibrium game (G) = 〈 , 〉 = A∪ D
G = ∪ strategy set S of the game is Vv x μ
strategy profile s is an element of S s = 〈 1,…, v , S1,… Sμ 〉∈ S
Network Games with Many Defenders
Profit ( ) individual profit of attacker , 1 ≤ i ≤v ( )=
individual profit of defender = |{ : ∃i, 1≤i ≤v , i }|
Network Games with Many Defenders
s is a Nash equilibrium if and only if
there exist D ⊂ and A ⊂ V which satisfy the following conditions: 1. 2. 3.
Network Games with Many Defenders
Theorem 2.3 If the number of attackers v is strictly less
than nV
then an edge model (G) has a Nash equilibrium
if and only if there exist D and A which satisfy
the following conditions:
Network Games with Many Defenders
Definition2.4 For a graph G, we have the following notations
max(G) denotes the game (G) where v = nV -m
and μ= nE
min(G) denotes the game (G) where v = m
and μ= nv-m
where m is the size of a maximum matching in G.
Network Games with Many Defenders
Definition 2.6 : a graph G is said to have the property Prop (*) if and only if for a minimum edge cover CE, there exists a map f : V {0,1} such that
for any multiple-edge star graph of CE with a center , = 0 Theorem 2.7: a game min(G) has a Nash equilibrium if and only
if G satisfies the property Prop (*) .
New Strategic Model
The new model is defined by interchanging the players’ roles.
Attackers Defenders
Original model
Attack a nodeof the network to damage
Protect the network by catching attackersin some part of the network
New model
Damage thenetwork by attacking an edge
Protect the network by choosing a vertex
New Strategic Model
A new strategic game α, δ(G) = 〈 , S 〉
on G is defined as follows S = Eα x Vδ is a strategy set of α, δ(G) s is an element of S s = 〈 e1,…, e α, v1,…,v δ 〉∈ S
Original model
New model
Game (G) = 〈 , 〉
α, δ(G) = 〈 ,
S 〉
S Vv x μ Eα x Vδ
New Strategic Model
Profit ( ) individual profit of attacker , 1≤ i ≤α
individual profit of defender , 1 ≤ j ≤ δ
New Strategic Model
Definition:
Theorem 3.2: |D| ≤ δ and |A| ≤ α where
New Strategic Model
Theorem 3.3 If α is the size of a maximum matching in G and δ=2α, then the game α,
δ(G) has a Nash
equilibrium.
and μ= nv-m
New Strategic Model
Definition 3.4 The graph G is bipartite if V=V0∪V1 for some disjoint vertex sets V0, V1 ⊆ V so that for each edge (u,v) ∈ E, u ∈ V0 and v ∈ V1
New Strategic Model
Theorem 3.5 For a bipartite graph G, a game α,
δ(G) has a
Nash equilibrium if and only if α, δ ≥ m, where m is the size of the maximum matching in G. Proof: For a bipartite graph G, if M is a maximum matching and is a minimum vertex cover, then such that
The End
Thanks for your attention.